Operator Theory and Free Harmonic Analysis

Hari Bercovici计划在两个基本方向上工作。第一个与沃库列斯库的自由概率论和谐波分析有关。这里的一个重要方向是研究乘法自由卷积的极限律。与经典调和分析不同，乘法卷积的研究不能简单地通过取对数来简化到加性情况。我们已经知道，有一些意想不到的“类泊松”概率分布与乘法自由卷积有关。一个重要的障碍是经典极限定理，甚至是中心极限定理，在自由乘法情况下的正确表述。Bercovici研究的第二条线是算子理论及其与函数理论、控制理论和其他领域的相互作用。该领域的具体问题包括算子的约束和压缩的模型理论，以及由算子交换族生成的对偶代数的研究。他还计划在相当容易理解的标量情况之外，重新研究倾斜Toeplitz算子。为了给这个提议中的研究提供一些动机，我想回顾一下，在目前对亚原子过程的描述中，人们必须处理概率过程。因此，例如，在任何给定时刻，粒子的位置都不能精确地知道。一个是粒子出现在给定空间区域的概率。一些版本的非交换概率论涉及量子化学的某些方面。自由概率论很可能出现在量子描述自然的某个点上。此外，它是一个有趣的研究对象，因为它在某种意义上提供了“唯一”的替代概率论。本课题的操作理论部分在很大程度上源于大型结构体控制理论提出的问题。该理论的一些早期结果被认为是NASP的主要控制机制。虽然Bercovici在这一理论的实践方面的专业知识有限，但控制理论家可以用理论术语来表述他们的问题，从而使算子理论方法的应用成为可能。

Abstract Bercovici Hari Bercovici plans to work in two basic directions. The first is related with Voiculescu's free probability theory and harmonic analysis. An important direction here is the sudy of limit laws for the multiplicative free convolution. Unlike classical harmonic analysis, the study of multiplicative convolution cannot simply be reduced to the additive case by taking logarithms. It is already known that there are some unexpected `Poisson like' probability distributions related with multiplicative free convolution. An important obstacle is the correct formulation of the classical limit theorems, even the central limit theorem, in the free multiplicative case. The second line of Bercovici's research is in operator theory and its interactions with function theory, control theory, and other areas. Among the specific problems in this area are the model theories of restrictions and compressions of operators, and the study of dual algebras generated by commuting families of operators. He also plans to return to the study of skew Toeplitz operators beyond the fairly well understood scalar case. In order to give some motivation for the research in this proposal I would like to recall that in the current description of subatomic processes one has to deal with probabilistic processes. Thus, for instance, the location of a particle is not known with precision at any given moment. One has instead the probability that that particle appears in a given region of space. Some versions of noncommutative probability theory deal with some of the aspects of quantum chemistry for instance. Free probability theory is likely to occur at some point in the quantum description of nature. Besides, it is an interesting object of study because it provides, in some sense, the `only' alternative probability theory. The operator theoretical part of this project stems to a great extent from problems raised by the control theory of large structures. Some earlier results in this theory were proposed a s the main control mechanism of the NASP. While Bercovici's expertise in the practical side of this theory is limited, control theorists can formulate their problems in theoretical terms, thus making the application of operator theoretical methods possible.